Dialetheia

A dialetheia is a truecontradiction - a statement such that both the statement and its negation are true. People who can stomach this idea, and believe there are such things, are called dialetheists - which doesn't imply anything about their religion: the word breaks down as di- (two) - aletheia (old Greek for truth - see aletheia where it is explained rather well).

A commonly given example is the 'liar paradox'
(Epimenides paradox, 'this sentence is false', etc.) The reasoning may go as follows: if it's true, it's false; if false then true; hence assuming either its truth or falsity implies both its truth and its falsity - it comes out the same.

Disregarding the falsities, both the statement and its contradiction may be said (by brave dialetheists, anyway) to be true.

For this idea to be any use in logic, we must use a deductive system where a true contradiction does not let you immediately prove all statements (ie. the Ex Falso Quodlibet rule, which is valid in the 'normal' propositional calculus, does not hold) otherwise the acceptance of one dialetheia would enable us to prove any
arbitrary statement, making our 'logic' trivial - any statement can be proved in it!